Claims

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1998_01_01_baker_harman: Corollary 1 of Baker and Harman's 1998 paper gives infinitely many n with more than n to the power 0.7039 preimages under Euler's totient, which settles the question for every epsilon at least 0.2961; refereed.

2022_11_14_lichtman: Corollary 1.3 of Lichtman's 2022 preprint gives infinitely many n with at least n to the power 0.7156 preimages under Euler's totient, which settles the question for every epsilon at least 0.2844; the site's best known bound.

2026_09_24_openai: The OpenAI release claims that for every positive epsilon infinitely many n have more than n to the power one minus epsilon preimages under Euler's totient, from a count of primes whose predecessor is x to the delta smooth.

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